Let be a measurable space. Let be a monotone set function (cf. also Set function) on , vanishing at the empty set, . Let be a non-negative measurable function and . The Choquet integral of on A with respect to is defined by
where the right-hand side is an improper integral and is the -cut of , [a1], [a2], [a6]. Specially, let be a simple measurable non-negative function on , , , and whenever . One can rewrite in the following form:
The Choquet integral has the following properties:
For any constant , .
If on , then .
For co-monotone functions and , i.e., for all , one has
Related integrals and generalizations.
Let be a non-negative extended real-valued measurable function on and . The Sugeno integral [a8] of on with respect to is defined by
where , .
|[a1]||G. Choquet, "Theory of capacities" Ann. Inst. Fourier (Grenoble) , 5 (1953) pp. 131–295|
|[a2]||D. Denneberg, "Non-additive measure and integral" , Kluwer Acad. Publ. (1994)|
|[a3]||M. Grabisch, H.T. Nguyen, E.A. Walker, "Fundamentals of uncertainity calculi with application to fuzzy inference" , Kluwer Acad. Publ. (1995)|
|[a4]||R. Mesiar, "Choquet-like integrals" J. Math. Anal. Appl. , 194 (1995) pp. 477–488|
|[a5]||T. Murofushi, M. Sugeno, "A theory of fuzzy measures. Representation, the Choquet integral and null sets" J. Math. Anal. Appl. , 159 (1991) pp. 532–549|
|[a6]||E. Pap, "Null-additive set functions" , Kluwer Acad. Publ. /Ister (1995)|
|[a7]||D. Schmeidler, "Integral representation without additivity" Proc. Amer. Math. Soc. , 97 (1986) pp. 253–261|
|[a8]||M. Sugeno, "Theory of fuzzy integrals and its applications" PhD Thesis Tokyo Inst. Technol. (1974)|
Choquet integral. E. Pap (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Choquet_integral&oldid=18610